Rin and Ryūji have never been rivals, not really.
Sure, there had been some friction between them when they first met, Ryūji's hardass diligent nature clashing with the half-demon's seemingly laid back and lazy behavior; not to mention the conscious effort put into starting shit on Ryūji's part that bordered on playground bullying. However, even during that time they could hardly be considered rivals, considering they never really competed with each other about anything, except perhaps the one time Ryūji drop kicked Rin during a running exercise because he couldn't stand the other 'winning' by being faster; an incident which, in essence, is a rather accurate summary of that period of Rin and Ryūji's relationship.
Their 'rivalry' was, truthfully, something Ryūji had constructed almost entirely on his own; it was a result of the image the boy had created of the other, and the roles he'd given to himself and Rin in his head, rather than their actual relationship. Ryūji was the hardworking, serious student with an ambition born from determination to keep his family together and take revenge from what caused it harm—– and Rin was the privileged little shit who got special treatment from teachers because of his connection to the headmaster, who slacked off and slept in class and didn't take anything seriously while people like Ryūji worked their asses off. It was dramatic and admittedly rather childish, especially considering that in hindsight, Rin had done nothing to encourage the other's behavior, and everything to get Ryūji to stop acting like an over-competitive fifth grader and leave him be. It had been a period of time that would make Ryūji cringe a little with embarrassment every time he recalled it, even if he'd learned from it and grown as a person.
Even that hadn't lasted long, however, Ryūji's abrasive and hostile attitude towards the other ceasing when Rin saved him both from a leaper Ryūji had aggravated out of pride and the heartbreak born of humiliation and despair. The image Ryūji had constructed of the guy –the infuriating, privileged slacker, the person who appeared to Ryūji as the embodiment of the people who got special treatment and whose mere presence mocked people like Ryūji who worked hard for their ambitions—was shatteredwhen Rin not only quite literally threw himself into the jaws of danger to save Ryūji, the person who had done nothing but insult and abuse him with no provocation whatsoever, but also announced the ambition he shared with Ryūji, boldly for everyone to hear; the way he scoffed it out in such a sure and matter-of-fact manner making the weight that had been crushing down on Ryūji for years due to everyone around him belittling his ambition dissipate in a single moment, a single heartbeat.
Ryūji would never admit it, not for years at least, but Rin had looked pretty damn cool at that moment—– he looked like a savior, like the hero that saved the damsel day in a movie, or perhaps the protagonist of a shōnen manga. That moment had been the first time Ryūji took his head out of his ass started actually seeing Rin for who he was rather than the image he had of him; it was the moment their very short-lived rivalry (which, admittedly, only existed on Ryūji's part) had ended, and their friendship began.
Once Ryūji stopped his less than pleasant behavior after the incident with the leaper he surprisingly ended up being the one to try to solve the enigma that was Rin more than anyone, poking and prodding and doing everything to understand his mystery of a friend who had both the world and himself convinced that he was nothing but an idiot when he was anything but. To Ryūji, Rin was a cluster of contradictions; he was an assortment of traits that Ryūji would have never thought would coexist in one single person, a puzzle that he needed to expand his mind to figure out. Rin was an idiot, and one of the smartest people Ryūji had ever met. He was brave, and so very fearful. He was tough as nails, nigh unbreakable—– and so fragile he could shatter like glass at the slightest touch. He was a roaring fire, bold, warm, burning, unashamed, unstoppable; he was a thick blanket of snow covering the world, suppressed, silent, muffled and hidden away from others.
He was the bastard of Satan, the offspring of the creature that had killed hundreds and made Ryūji's home into the cursed temple, the child of the king of all demons and incarnation of Gehenna itself—– and that didn't matter one bit to Suguro Ryūji, to whom Rin was simply an aggressively caring, kind, self-sacrificial dumbass who carried any burden he could by himself and pissed Ryūji off with that bullshit. He was Okumura Rin, the most loving person Ryūji had ever met, his heart bigger than his body and his dedication to those he loved bigger than both of them combined, the boy who would throw himself into the jaws of danger and an aggravated leaper for someone who had been nothing but a grade-A asshat to him.
It was funny if not somewhat poetic, really; the fact that Ryūji, who couldn't see Rin past the image he'd created of the boy at first, ended up being the one to see past the image Rin had created of himself—– and that the person Ryūji thought he hated ended up being the one he cherished and loved so much his place in Ryūji's heart, life and very soul is one that no other could share or replace.